Monomial

In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: A monomial, also called a power product or primitive monomial, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, x 2 y z 3 = x x y z z z {\displaystyle x^{2}yz^{3}=xxyzzz} is a monomial. The constant 1 {\displaystyle 1} is a primitive monomial, being equal to the empty product and to x 0 {\displaystyle x^{0}} any variable x {\displaystyle x} . If only a single variable x {\displaystyle x} is considered, this means that a monomial is either 1 {\displaystyle 1} or a power x n {\displaystyle x^{n}} of x {\displaystyle x} , with n {\displaystyle n} a positive integer. If several variables are considered, say, x , y , z , {\displaystyle x,y,z,} then each can be given an exponent so that any monomial is of the form x a y b z c {\displaystyle x^{a}y^{b}z^{c}} with a , b , c {\displaystyle a,b,c} non-negative integers (taking note that any exponent 0 {\displaystyle 0} makes the corresponding factor equal to 1 {\displaystyle 1} ). A monomial in the first sense is multiplied by a nonzero constant, called the coefficient of the monomial. A primitive monomial is a special case of a monomial in this second sense, where the coefficient is 1 {\displaystyle 1} . For example, in this interpretation, − 7 x 5 {\displaystyle -7x^{5}} and ( 3 − 4 i ) x 4 y z 13 {\displaystyle (3-4i)x^{4}yz^{13}} are monomials (in the second example, the variables are x , y , z , {\displaystyle x,y,z,} and the coefficient is a complex number). In the context of Laurent polynomials and...